1. **State the problem:** We need to find the integral $$\int (x^2 + 3x)^2 \, dx$$. 2. **Use the formula:** To integrate a polynomial squared, first expand the expression inside the integral. 3. **Expand the square:** $$ (x^2 + 3x)^2 = (x^2)^2 + 2 \cdot x^2 \cdot 3x + (3x)^2 = x^4 + 6x^3 + 9x^2 $$ 4. **Rewrite the integral:** $$ \int (x^4 + 6x^3 + 9x^2) \, dx $$ 5. **Integrate term-by-term:** - $$\int x^4 \, dx = \frac{x^5}{5}$$ - $$\int 6x^3 \, dx = 6 \cdot \frac{x^4}{4} = \frac{3x^4}{2}$$ - $$\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3$$ 6. **Combine the results:** $$ \frac{x^5}{5} + \frac{3x^4}{2} + 3x^3 + C $$ 7. **Final answer:** $$ \int (x^2 + 3x)^2 \, dx = \frac{x^5}{5} + \frac{3x^4}{2} + 3x^3 + C $$ Where $C$ is the constant of integration.